You have uploaded a question bank for Engineering Mathematics-I (Periodic Test-I) from MGM's College of Engineering. It covers a variety of topics, primarily focusing on matrix operations, including:

1. Matrix Rank: Finding the rank by converting matrices into Echelon or Normal form.
2. Matrix Inversion: Using the Gauss-Jordan method to find the inverse of matrices.
3. System of Linear Equations: Testing consistency and solving systems of equations.
4. Eigenvalues and Eigenvectors: Finding these for various matrices.
5. Cayley-Hamilton Theorem: Verification and application to find inverses.
6. Partial Derivatives: Finding first and second order derivatives.
7. Advanced Mathematical Proofs: Proof-based problems involving partial derivatives and vector calculus.

Let me know if you need help with any specific question from this document, and I'd be happy to assist!

Would you like to focus on a particular problem? Here are some examples to consider:

1. Solve for matrix rank from the question Q1 or Q2.
2. Find the inverse of a matrix using the Gauss-Jordan method from Q3.
3. Analyze the consistency of a system of equations from Q4.
4. Find eigenvalues and eigenvectors from Q7.
5. Solve a partial derivative problem from Q11.

Tip: When solving for matrix rank, remember that it's the number of non-zero rows after converting a matrix into row echelon form.